• CfQ 
394 
me. a. CAYLEY ON THE CONIC OF FIVE-POINTIC 
ential, is a denYative point of the degree (i+lf, and it is easy to see that the degree 
IS not 9 ; it is therefore 25. The point of simple intersection of the five-pomtic come is 
a derivatire point of the degree 26 ; it is therefore, according to Professor Sylthstees 
general theory, identical with the point given by the foimer constniction ; this agrees 
with the before-mentioned theorem of Mr. SalmoisL 
IV. Independent investigation for the Cubic. 
47. The following is, in substance, the method by which I fii’st obtained the equation 
of the five-pointic conic, for the cubic 
Z®+6^XYZ=0. 
Write for shortness 
V = + 2 + (/ + 2 lzx)Y + ( 2 " + 2 , 
W=(X^+2WZ)^-1-(Y"+2^Z)2/-1-(Z"+2/XY)2, 
T =X®+Y*-1-Z"+6ZXYZ, 
P -^ax-Yby-^-cz, 
n=aX+5Y+cZ. 
Then X, Y, Z being current coordinates, and x, y, z the coordinates of a point of the 
cubic (so that U = 0), the equation of the cubic will be 
T=0, 
and the equation of a conic having with it an ordinary (two-pointic) contact at the point 
(ayy, z), wdlbe* 2W— nV=0. 
48 Now imagine from the point of contact lines drawn to the other four mtersec- 
tions of the two curves; in the case of the five-pointic conic, three of these lines wil 
coincide with the tangent V=0, and the remaining line wall be the line joining the pom 
of contact with the point of simple intersection. The equations of the lines in question 
can be found by Joachimsthal’s theorem, viz. if ( x , J, z) be rtie coordinates of a giien 
noint and (X, Y, Z) current coordinates, then if in the equations of any two curies we 
substitute for the coordinates, X*-f^X, Xy-(-^Y, Xz-t-fiZ, and between the equations so 
obtained eliminate X, fz, the resulting equation will be that of the lines iaini fioni le 
point (a-, ?/, z) to the points of intersection of the two cuives. The point (a, y, z) is an; 
point whatever, and it may therefore be a point of intersection, or, as in the piesen , 
instance, a point of contact of the two curves ; the only difference is, that m either case 
the degree of each equation as regards (X, ^ is reduced by unity, and the degree of the 
resulting equation in X, Y, Z is also reduced by unity : in the case of a point of simple 
intersection this is the only reduction; but in the case of a point ot contact, the result- 
* I have introduced the factor 2, to make this correspond with the form D'^U-nDU^O, in the case in 
Lj[uestiou, m—2>. 
